A presentation from the Telemetrics Lab

Transcription

1 A presentation from the Telemetrics Lab Telemetrics lab Department of Psychology Northwestern University Evanston, Illinois USA November, 2010

2 Outline 1 Data from a Correlation Matrix Simulated data Real data Ability tests Factor diagrams Orthogonal Rotations 2 Raw data From a built in data set 3 Alternatives to Factor Analysis Hierarchical Cluster Analysis 4 Data from an external file

3 Introduction Factor analysis several examples Data from a correlation matrix Simulated 2 factor data Real data Ability tests Raw data Simulated 2 factor data Real data 5 Personality dimensions

4 Simulated data Simulate 2 factor data Using the sim.item function > set.seed(42) #to generate a reproducible example > my.data <- sim.item(12) > my.cor <- cor(my.data) > round(my.cor,2) V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V V V V V V V V V V V V

5 Simulated data How many factors in my.cor > fa.parallel(my.cor,n.obs=500) Parallel analysis suggests that the number of factors = 2 and the number of components = 2 Parallel Analysis Scree Plots eigenvalues of principal components and factor analysis PC Actual Data PC Simulated Data FA Actual Data FA Simulated Data

6 Simulated data Try Very Simple Structure as well as MAP > vss(my.cor,n.obs=500) Very Simple Structure Call: VSS(x = x, n = n, rotate = rotate, diagonal = diagonal, fm = fm, n.obs = n.obs, plot = plot, title = title) VSS complexity 1 achieves a maximimum of 0.74 with 3 factors VSS complexity 2 achieves a maximimum of 0.8 with 8 factors The Velicer MAP criterion achieves a minimum of 0.02 with 2 factors Velicer MAP [1] Very Simple Structure Complexity 1 [1] Very Simple Structure Complexity 2 [1]

7 Simulated data Examine the output Very Simple Structure Very Simple Structure Fit

8 Simulated data Extract 2 factors part 1 > fa(my.cor,2,n.obs=500) Factor Analysis using method = minres Call: fa(r = my.cor, nfactors = 2, n.obs = 500) Standardized loadings based upon correlation matrix MR1 MR2 h2 u2 V V V V V V V V V V V V MR1 MR2 SS loadings Proportion Var Cumulative Var

9 Simulated data 2 artificial factors part 2 With factor correlations of MR1 MR2 MR MR Test of the hypothesis that 2 factors are sufficient. The degrees of freedom for the null model are 66 and the objective function wa The degrees of freedom for the model are 43 and the objective function was 0.1 The root mean square of the residuals is 0.02 The df corrected root mean square of the residuals is 0.03 The number of observations was 500 with Chi Square = with prob < 0.11 Tucker Lewis Index of factoring reliability = RMSEA index = and the 90 % confidence intervals are BIC = Fit based upon off diagonal values = 0.99 Measures of factor score adequacy MR1 MR2 Correlation of scores with factors Multiple R square of scores with factors Minimum correlation of possible factor scores

10 Factor Number Real data Ability tests 9 mental tests from Thurstone data(bifactor) fa.parallel(thurstone,n.obs=213) Parallel Analysis Scree Plots eigenvalues of principal components and factor analysis PC Actual Data PC Simulated Data FA Actual Data FA Simulated Data

11 Real data Ability tests Exract 3 factors > fa3 <- fa(thurstone,3,n.obs=213) > fa3 Factor Analysis using method = minres Call: fa(r = Thurstone, nfactors = 3, n.obs = 213) Standardized loadings based upon correlation matrix MR1 MR2 MR3 h2 u2 Sentences Vocabulary Sent.Completion First.Letters Letter.Words Suffixes Letter.Series Pedigrees Letter.Group MR1 MR2 MR3 SS loadings Proportion Var Cumulative Var

12 Real data Ability tests Thurstone 3 factors part 2 With factor correlations of MR1 MR2 MR3 MR MR MR Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 36 and the objective function wa The degrees of freedom for the model are 12 and the objective function was 0.0 The root mean square of the residuals is 0 The df corrected root mean square of the residuals is 0.01 The number of observations was 213 with Chi Square = 2.82 with prob < 1 Tucker Lewis Index of factoring reliability = RMSEA index = 0 and the 90 % confidence intervals are BIC = Fit based upon off diagonal values = 1 Measures of factor score adequacy MR1 MR2 MR3 Correlation of scores with factors Multiple R square of scores with factors

13 Factor diagrams A factor diagram fa3 <- fa(thurstone,3,n.obs=213) Factor Analysis Sentences Vocabulary Sent.Completion MR1 First.Letters 4.Letter.Words Suffixes MR Letter.Series Letter.Group MR3 Pedigrees

14 Orthogonal Rotations Thurstone, 3 factors Varimax rotated > v3 <- fa(thurstone,3,rotate="varimax",n.obs=213) > fa.diagram(v3) > v3 Factor Analysis using method = minres Call: fa(r = Thurstone, nfactors = 3, n.obs = 213, rotate = "Varimax") Standardized loadings based upon correlation matrix MR1 MR2 MR3 h2 u2 Sentences Vocabulary Sent.Completion First.Letters Letter.Words Suffixes Letter.Series Pedigrees Letter.Group MR1 MR2 MR3 SS loadings Proportion Var Cumulative Var

15 > fa.diagram(v3) Orthogonal Rotations Compare the two solutions > v3 <- fa(thurstone,3,rotate="varimax",n.obs=213) > fa.diagram(v3) Factor Analysis Factor Analysis Sentences Sentences Vocabulary Sent.Completion First.Letters 4.Letter.Words Suffixes Letter.Series Letter.Group MR1 MR2 MR3 Vocabulary Sent.Completion First.Letters 4.Letter.Words Suffixes Letter.Series Letter.Group Pedigrees MR1 MR2 MR Pedigrees

16 From a built in data set R has many built in data sets data(bfi) 25 personality items from the Big 5 Collected as part of the SAPA project Thought to represent 5 dimensions Agreeableness Extraversion Conscientiousness Extraversion Neuroticism

17 From a built in data set Describe the Big 5 > data(bfi) > describe(bfi) var n mean sd median trimmed mad min max range skew kurtosis se A A A A A C C C C C E E E E E N N N N N O O O O O gender education age

18 From a built in data set How many factors? > fa.parallel(bfi[1:25]) #just the items Parallel analysis suggests that the number of factors = 6 and the number of co Parallel Analysis Scree Plots eigenvalues of principal components and factor analysis PC Actual Data PC Simulated Data PC Resampled Data FA Actual Data FA Simulated Data FA Resampled Data Factor Number

19 From a built in data set How many factors part 2: VSS > VSS(bfi[1:25]) Very Simple Structure Call: VSS(x = bfi[1:25]) VSS complexity 1 achieves a maximimum of 0.58 with 4 factors VSS complexity 2 achieves a maximimum of 0.74 with 4 factors The Velicer MAP criterion achieves a minimum of 0.01 with 5 factors Velicer MAP [1] Very Simple Structure Complexity 1 [1] Very Simple Structure Complexity 2 [1]

20 From a built in data set VSS plot Very Simple Structure Very Simple Structure Fit

21 From a built in data set Extract 5 factors from the BFI > f5 <- fa(bfi[1:25],5) fa.diagram(f5,main="five factors of personality?") Five factors of personality? N1 N2 N3 N5 C4 C2 C5 C3 C1 A3 A2 A5 A4 A1 E2 E1 E4 N4 E5 E3 O3 O1 O5 O2 O MR2 MR3 MR5-0.3 MR1 MR4

22 Hierarchical Cluster Analysis ICLUST of Big 5 > iclust(bfi[1:25]) ICLUST (Item Cluster Analysis Purified Alpha: C20 C16 C15 C G6* reliability: C20 C16 C15 C Original Beta: C20 C16 C15 C Cluster size: C20 C16 C15 C Purified scale intercorrelatio correlations corrected for att reliabilities on diagonal C20 C16 C15 C21 C C C C With eigenvalues of: C20 C16 C15 C

23 Hierarchical Cluster Analysis ICLUST as a graphic tree structure Hierarchical Clusters of the Big 5 O5 O2 O4 O3 O1 E5 E3 E4 E2 E1 A5 A3 A2 A4 A1 N2 N1 N5 N4 N3 C5 C4 C3 C2 C C C C C C5 C C C8 C C11 α = 0.72 β = C12 α = 8 β = C10 α = 0.72 β = C16 C13 α = β = 0.76 α = 0.71 β = 5 C15 C14 α = β = 7 α = 3 β = 0.58 C19 α = 0.76 β = 4 C17 9 α = 0.72 C18 β = α = 0.71 β = C20 α = 0.81 β = 3 C21 α = 0.41 β = 0.27

24 Analyzing from an external file Data may reside on a local or a remote computer Option A: Using read.clipboard and its alternatives Open the other other file using a text editor or spreadsheet program Select all and copy (to the clipboard) my.data <- read.clipboard() or my.data <- read.clipboard.csv() or read.clipboard.tab() Read the information directly find the file and call it something fn <- file.choose() Read in the data my.data <- read.table(fn, header=true) Read from an SPSS file using the foreign package library(foreign) find the file and call it something fn <- file.choose() my.data <- read.spss(fn,to.data.frame=true)